Why is the Blender soft body physics permanent deformation not working?
In order for the Soft Body 'plastic' setting to have a meaningful effect you need to disable the 'goal' for those vertices - otherwise the 'goal' will simply pull the vertices back to their original positions.

However, in your situation it probably isn't as simple as disabling the 'goal' since the soft body will then be subject to other forces such as gravity so you'll need to create a Vertex Group of just those vertices that would not be expected to move (eg, the wheels) that the rest of the mesh can be supported upon. In the Soft Body Goal settings you can then leave the 'goal' enabled and set the Vertex Group to limit the goal to only the desired vertices (eg, the wheels that would be resting on the floor).

This can produce the following :